A sequence where each term is multiplied by the same number, to get the next term is a geometric sequence. The number can be any non-zero real number. In the following example, is making each term twice as large as the previous.
Similar to other sequences, the first term is usually called the second and so on.
Consider the following geometric sequence. Determine the common ratio and find the next three terms.
In geometric sequences, the terms increase or decrease by a common ratio. Since we know that this sequence is geometric, it's enough to find the ratio between two consecutive terms. The ratio for the others must then be the same. Let's take the first two: If we let be the common ratio we get the equation The common ratio, is To find the next terms, we multiply by three times. In summary, the common ratio is and the next three terms are and
This is very similar to the rate of change of an exponential function. The number would, in that case, be the constant multiplier. In fact, when graphing a geometric sequence in a coordinate plane, it resembles the graph of an exponential function.
A vet gives medicine to an axolotl for a week. The first dose is mg, and every day it's cut in half. List the doses in a sequence and graph it in a coordinate plane.
The first dose is mg, so that is the first term. The next dose is half of that. The second term is mg. We can find the rest of the terms by continuing to multiply by The sequence is If we let be the days and be the dose in mg we can graph the sequence.